Title: Properties of Kac-Moody Nilmanifolds and Solvmanifolds
Abstract:We explore the geometric properties of Kac-Moody
nilmanifolds and solvmanifolds. These are defined by Kac-Moody
algebras, and their solvable subalgebras can be viewed as
generalizations of noncompact symmetric spaces. While Kac-Moody
algebras are infinite dimensional, their solvable subalgebras have
finite dimensional quotients.
The Ricci curvature form of a submanifold is not, in general, the
restriction of the Ricci curvature of the ambient space. Therefore,
classes of manifolds on which the Ricci curvatures of some
submanifolds are aligned are very special. Indeed, Tamaru exploited
this idea in the setting of noncompact symmetric spaces to construct
new examples of Einstein solvmanifolds via special subalgebras. We
characterize the largest category in which Tamaru's construction can
be extended, identifying two crucial algebraic/metric conditions. On
Kac-Moody solvmanifolds, our crucial extra conditions hold. In current
work in progress, we investigate other geometric properties of these
spaces.
This is joint work with Tracy Payne (Idaho State University).